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Math Help - Lebesgue measure on sigma-fields of Borel sets

  1. #1
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    Lebesgue measure on sigma-fields of Borel sets

    Hello, I am completely lost with this problem. Any guidance is greatly appreciated.

    Let \Omega=\{(x,y):0<x\leq 1,0<y\leq 1\}, let \mathcal{B}^2 be the \sigma-field of Borel subsets of \Omega and let P=\lambda_2 be the Lebesgue measure on \mathcal{B}^2. Let A_0=\{(x,y):x>0,y>0,x+y\leq 1\}.

    a. Starting from the fact that \mathcal{B}^2=\sigma(\mathcal{R}^{0,2}), show that A_0 \in \mathcal{B}^2, where \mathcal{R}^{0,2} is the collection of two-dimensional bounded rectangles of the form  B(a_1,b_1,a_2,b_2) = \{(x,y):a_1<x \leq b_1, a_2< y \leq b_2\} with a_1,b_1,a_2,b_2 \in [0,1], and a_1 \leq b_1, a_2 \leq b_2


    b. Let  P_* and P^* be the inner and outer measures defined from  P=\lambda_2. Prove that P_*(A_0)=P^*(A_0)=P(A_0)=1/2. (Hint: one way to do this is to construct sequences of sets B_n and C_n of known measure , e.g. unions of disjoint rectangles, with B_n \downarrow A_0 and C_n \uparrow A_0)

    Thanks in advance
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: Lebesgue measure on sigma-fields of Borel sets

    Quote Originally Posted by akolman View Post
    Hello, I am completely lost with this problem. Any guidance is greatly appreciated.

    Let \Omega=\{(x,y):0<x\leq 1,0<y\leq 1\}, let \mathcal{B}^2 be the \sigma-field of Borel subsets of \Omega and let P=\lambda_2 be the Lebesgue measure on \mathcal{B}^2. Let A_0=\{(x,y):x>0,y>0,x+y\leq 1\}.

    a. Starting from the fact that \mathcal{B}^2=\sigma(\mathcal{R}^{0,2}), show that A_0 \in \mathcal{B}^2, where \mathcal{R}^{0,2} is the collection of two-dimensional bounded rectangles of the form  B(a_1,b_1,a_2,b_2) = \{(x,y):a_1<x \leq b_1, a_2< y \leq b_2\} with a_1,b_1,a_2,b_2 \in [0,1], and a_1 \leq b_1, a_2 \leq b_2
    Each of those rectangles is borel....soo.

    b. Let  P_* and P^* be the inner and outer measures defined from  P=\lambda_2. Prove that P_*(A_0)=P^*(A_0)=P(A_0)=1/2. (Hint: one way to do this is to construct sequences of sets B_n and C_n of known measure , e.g. unions of disjoint rectangles, with B_n \downarrow A_0 and C_n \uparrow A_0)

    Thanks in advance
    I mean, do you have any idea?
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  3. #3
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    Re: Lebesgue measure on sigma-fields of Borel sets

    I have an idea, but there are some gaps I'm not really sure how to fill in.

    Let B_n = \cup_{i=1}^N\{(x,y):\frac{i-1}{2^n} < x \leq \frac{i}{2^n}, 0 < y \leq 1- \frac{i-1}{2^n}\}
    By construction  B_{n+1} \subset B_n and  A_0 \subset \cap_{n=1}^\infty B_n.
    I am pretty sure that \cap_{n=1}^\infty B_n \subset A_0 but I don't know how to prove it.

    For the second part, if the above is true B_n \downarrow A_0 so P(B_n) \downarrow P(A_0) which should be 1/2. since P=\lambda_2 and \mathcal{B}^2 is a \sigma field the outer and inner measure should agree.

    Is that on the right track? any hints to fill in the gaps? thanks again.
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